# DM865 @ SDU - Heuristics and Approximation Algorithms

#### Sheet 1:

1. Let $G$ be a complete undirected graph with non-negative edge weights.

a) Let $W$ denote the maximum weight of any edge in $G$.
For each edge $e$, add $W$ to the weight of $e$.
Let $G’$ denote the resulting weighted graph.

Prove that the weights of $G’$ are metric, i.e., prove that they satisfy the triangle inequality.

b) Argue that a TSP tour in $G$ is optimal, if and only if the corresponding tour in $G’$ is optimal.

c) Why doesn’t this contradict Theorem 2.9?

2. Argue that (the decision version of) metric TSP is NP-hard.

3. Describe an algorithm for finding an Euler tour in a graph where all vertices have even degree.

4. a) Give an example where Christofide’s Algorithm produces a better solution than the Double Tree Algorithm. More specifically, give a graph and concrete runs of the two algorithms on the graph such that Christofide’s Algorithm produces a cycle of smaller total weight than the Double Tree Algorithm.

b) Do a) again, but such that the Double Tree Algorithm gives the better result.

c) How many nodes do you need to construct the examples in a) and b)?

#### Sheet 2:

1. Read the Python tutorial [No]. You find some starting code from that page here.

2. Implement the exact methods: plain enumeration and Held Karp dynamic programming algorithm.

3. Following the procedure for Benchmarking described in [No] implement and compare as many TSP heuristics as you can. You find a list below, in bold the heuristics implemented in [No]. For a description of these heuristics see [Be].

• Heuristics that Grow Fragments
• Nearest neighborhood heuristic
• Double-Ended Nearest Neighbor heuristic
• Multiple Fragment heuristic (aka, greedy heuristic)
• Heuristics that Grow Tours
• Clarke-Wright savings heuristic
• Nearest Insertion
• Farthest Insertion
• Random Insertion
• Heuristics based on Trees
• Minimum spanning tree heuristic
• Christofides’ heuristics
• Fast recursive partitioning heuristic
4. In Python kd-trees are already implemented in the module scipy. Try to use them and improve some implementations of the construction heuristics described above (you will probably need to change the representation of points).

#### Sheet 3:

1. In the code available in the git repository you find a file local_search.py, which contains an implementation of a 2 opt local search. Study the implementation and test the results when the local search is executed after different construction heuristics. Is the local search implemented in that file a first improvement or a best improvement? Does the 2-opt algorithm improve the results of the construction heuristics? How many steps (changes in the solutions) are executed? Which combination construction_heuristic + 2_opt leads to the best results (including a random initial solution)?

2. Compare the results of a first improvement 2-opt against those of a best improvement 2-opt procedure. Is the comparison the same across different initial solutions attained by different construction heuristics (including a random solution and a canonical sequence)?

3. Try to improve the 2-opt implementation from the previous point. Start by adding random choices. Then, improve its execution time by adopting some of the techniques explained in class.

4. Consider the traveling salesman problem defined on an incomplete graph. How could we encode the problem such that we can approach it with the construction heuristics and local search algorithms implemented for the complete graph version of the problem?

5. Consider the asymmetric TSP. How can we encode this problem into a symmetric TSP, such that we can approach it with the construction heuristics and local search algorithms implemented for the symmetric version of the problem?

#### Sheet 4:

Design a 3 exchange iterative improvement procedure for the TSP. The procedure must return a local optimum in the 3 exchange neighborhood. Implement the procedure in the framework made available in git. A template to be completed is available in the file three_opt.py.

You must only edit this file, you are not allowed to change the other files. When executed, your program will read the instance USA, construct a canonical tour and call your iterative improvement procedure. The benchmarking called from the main file will take care of assessing the quality of your solution.

Describe your algorithm in pseudocode in a one-page document edited with Latex. Use the Latex package algorithm2e.

Submit only the file three_opt.py and the PDF result of your Latex pseudocode at this portal. Keep your files anonymous!

You are encouraged to work in pairs at this assignment, in which case it is enough that only one submits.

Remember: start out with simple and even inefficient code without optimizing for efficiency. Only later, when your initial implementation is working and doing what you expect, start looking at efficiency improvements of your code (and consider the quality of the solutions as well.

Instructions for the submission to http://valkyrien.imada.sdu.dk/DOApp/:

• Submit separately the source code (tgz file) and the description (pdf file).

• Organize the source file like it is organized in the git repository from where you got the starting package. Create the archive from the root of the repository (that is, the directory that contains src/) using the following command:

tar czvf TSP.tgz * --exclude=.git --exclude=__*


Include a Makefile in the src directory. It can be empty.

#### Sheet 5:

Argue that $\text{Rand}_p$ with $p=\frac{1}{2}(\sqrt{5}-1)$ (the algorithm from Section 5.3) can be derandomized to obtain a deterministic $p$-approximation algorithm.

#### Sheet 6:

1. a) Write down an LP-formulation of the unweighted Vertex Cover problem.

b) Write down the dual of the LP from a).

c) Which combinatorial problem does the dual correspond to?

2. Although the unweighted Vertex Cover problem is NP-hard for general graphs, there are graph classes that allow for efficient algorithms. Design an algorithm that finds a minimum cardinality vertex cover of a tree in linear time.

3. a) Assume that you have an algorithm for finding a minimum cardinality vertex cover in a graph.
Explain how you can use the algorithm for finding a maximum cardinality independent set.

b) Does this mean that you can use an approximation algorithm for unweighted Vertex Cover, like the ones in Sections 1.3 and 1.4, for approximating a maximum cardinality independent set? (Hint: What approximation factor could you obtain?)

#### Sheet 7:

1. Consider the primal-dual algorithm for the unweighted Vertex Cover problem.

a) What does the algorithm do?

b) Write down the same algorithm without explicitly using the LP-formulation of the problem.

c) Give an example showing that the algorithm has an approximation factor of at least 2.

2. Do Exercise 5.7. Hint: Using $\lambda = n \cdot \ln n \cdot Z_{\text{LP}}^*$, it is possible to obtain a $(3 \ln n)$-approximation algorithm.

#### Sheet 8:

1. Solve Exercise 3.1.

Hint: For proving the appoximation ratio it may be helpful to first consider the algorithm that chooses between the sets {1,2,…,k} and {k+1}.

#### Sheet 9:

Classfy the following scheduling applications:

1. Gate Assignment at an Airport

• Airline terminal at a airport with dozes of gates and hundreds of arrivals each day.

• Gates and Airplanes have different characteristics

• Airplanes follow a certain schedule

• During the time the plane occupies a gate, it must go through a series of operations

• There is a scheduled departure time (due date)

• Performance measured in terms of on time departures.

2. Scheduling Tasks in a Central Processing Unit (CPU)

• Schedule time that the CPU devotes to the different programs

• Exact processing time unknown but an expected value might be known

• Each program has a certain priority level

• Tasks are often sliced into little pieces. They are then rotated such that low priority tasks of short duration do not stay for ever in the system.

• Minimize expected time %, ie, sum of the weighted completion times for all tasks

3. Paper bag factory

• Basic raw material for such an operation are rolls of paper.

• Production process consists of three stages: printing of the logo, gluing of the side of the bag, sewing of one end or both ends.

• Each stage consists of a number of machines which are not necessarily identical.

• Each production order indicates a given quantity of a specific bag that has to be produced and shipped by a committed shipping date or due date.

• Processing times for the different operations are proportional to the number of bags ordered.

• There are setup times when switching over different types of bags (colors, sizes) that depend on the similarities between the two consecutive orders

• A late delivery implies a penalty that depends on the importance of the order or the client and the tardiness of the delivery.

#### Sheet 10:

1. Solve Exercise 2.2

2. In the lecture on Tuesday, May 21, we proved that the approximation ratio of the List Scheduling algorithm is at most $2−\frac{1}{m}$. Prove that this bound is tight, i.e., prove that the ratio is at least $2−\frac{1}{m}$.